Question

question
an element with a mass of 240 grams decays by 17.4% per minute. to the nearest tenth of a minute, how long will it be until there are 20 grams of the element remaining?
answer attempt 1 out of 2

Explanation:

Step1: Set up the decay formula

The formula for exponential decay is $A = A_0(1 - r)^t$, where $A$ is the final amount, $A_0$ is the initial amount, $r$ is the rate of decay, and $t$ is the time. Here, $A_0=240$, $r = 0.174$, and $A = 20$. So the equation becomes $20=240(1 - 0.174)^t$, which simplifies to $20 = 240\times0.826^t$.

Step2: Simplify the equation

Divide both sides of the equation by 240: $\frac{20}{240}=0.826^t$, so $0.0833\approx0.826^t$.

Step3: Take the natural - logarithm of both sides

$\ln(0.0833)=\ln(0.826^t)$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we get $\ln(0.0833)=t\ln(0.826)$.

Step4: Solve for $t$

$t=\frac{\ln(0.0833)}{\ln(0.826)}$. We know that $\ln(0.0833)\approx - 2.4849$ and $\ln(0.826)\approx - 0.191$. Then $t=\frac{- 2.4849}{- 0.191}\approx13.5$.

Answer:

13.5